Problem 53 - Elementary Algebra

Question

For the following problems, convert the numbers from scientific notation to standard decimal form. Light travels about \(5.866 \times 10^{12}\) miles in one year.

Step-by-Step Solution

Answer

Question: Convert the given scientific notation, \(5.866 \times 10^{12}\), which represents the distance light travels per year in miles, to standard decimal form. Answer: \(5,866,000,000,000\) miles

Step 1: Identify the parts of scientific notation

In the given scientific notation, \(5.866 \times 10^{12}\), there are two parts: 1. The coefficient: \(5.866\) 2. The base and exponent: \(10^{12}\) To convert to standard decimal form, we simply multiply the coefficient with \(10\) raised to the power of the exponent.

Step 2: Multiply the coefficient by the base raised to the power of the exponent

In this step, first calculate the base (\(10\)) raised to the power of the exponent (\(12\)) and then multiply it with the coefficient (\(5.866\)). \(10^{12} = 1,000,000,000,000\) Now multiply this value with the coefficient: \(5.866 \times 1,000,000,000,000 = 5,866,000,000,000\)

Step 3: Write the final result in standard decimal form

The given value in scientific notation, \(5.866 \times 10^{12}\), can be written in standard decimal form as: \(5,866,000,000,000\) So, light travels about \(5,866,000,000,000\) miles in one year.

Key Concepts

Decimal Form

The decimal form is the representation of numbers as they are commonly shown, without exponents or powers. It is the standard way of writing numbers down so everyone understands them immediately.
For example, the number known as "ten million" is written as 10,000,000 in decimal form. Converting from scientific notation to decimal form involves understanding each part included in the scientific notation, such as the coefficient and the exponent, as well as the base, typically 10.
In scientific notation, the process to convert to decimal form involves moving the decimal point in the coefficient as many places as indicated by the exponent.
For instance, the scientific notation \(5.866 \times 10^{12}\) requires moving the decimal in 5.866 twelve places to the right, resulting in the decimal form \(5,866,000,000,000\). This highlights the power of scientific notation in simplifying very large numbers.

Coefficient

The coefficient is a crucial part of scientific notation. It is the number that you'll find multiplying the base and its exponent. In the expression \(5.866 \times 10^{12}\), the coefficient is 5.866.
Understanding the coefficient is important because it helps define the significant figures of the number. It indicates the precision level of the expression.

• The coefficient is always a number greater than or equal to 1 but less than 10.
• It’s what we multiply by the power of 10 to get a full, clear number in decimal form.

When you multiply the coefficient by the base raised to the exponent, you scale the number up to its true value, maintaining the accuracy that the coefficient provides.

Exponent

An exponent in scientific notation represents how many times the base should be multiplied by itself. In our example of \(5.866 \times 10^{12}\), the exponent is 12.
Exponents are especially useful for very large or very small numbers, making them easier to write and work with.

• Positive exponents, like 12, indicate a large number and mean you shift the decimal point to the right in the coefficient.
• Negative exponents imply a small number, moving the decimal to the left.

In this way, exponents help us convert from scientific notation to decimal form without calculating huge numbers directly, by simply understanding the scale indicated by the exponent.

Base 10

Base 10 is the foundation of our positional number system, also known as the decimal system. In scientific notation, this "10" is used as the base to express numbers compactly.
Why base 10? It’s because it's convenient; our usual number system uses ten unique digits (0 through 9). In scientific notation, large numbers are simplified by expressing them as multiples of 10 raised to various powers.

• The number 10 itself signifies a complete set of ten digits, which is why it fits perfectly in our standard number system.
• When raised to any power, it represents the place value change—teacher.” for example, units, tens, hundreds, and so on.

Understanding base 10 is vital because it’s the backbone of how numbers expand and shrink in scientific notation, supporting seamless translation between scientific and decimal forms.